Editing Total variation (section)

== Applications ==
Total variation can be seen as a [[non-negative]] [[real number|real]]-valued [[functional (mathematics)|functional]] defined on the space of [[real number|real-valued]] [[function (mathematics)|function]]s (for the case of functions of one variable) or on the space of [[integrable function]]s (for the case of functions of several variables). As a functional, total variation finds applications in several branches of mathematics and engineering, like [[optimal control]], [[numerical analysis]], and [[calculus of variations]], where the solution to a certain problem has to [[Maxima and minima|minimize]] its value. As an example, use of the total variation functional is common in the following two kind of problems

* '''Numerical analysis of differential equations''': it is the science of finding approximate solutions to [[differential equation]]s. Applications of total variation to these problems are detailed in the article "''[[total variation diminishing]]''"
* '''Image denoising''':<ref>https://arxiv.org/pdf/1603.09599 Retrieved 12/15/2024</ref> in [[image processing]], denoising is a collection of methods used to reduce the [[Electronic noise|noise]] in an [[image]] reconstructed from data obtained by electronic means, for example [[data transmission]] or [[Sensor|sensing]]. "''[[Total variation denoising]]''" is the name for the application of total variation to image noise reduction; further details can be found in the papers of {{Harv|Rudin|Osher|Fatemi|1992}} and {{Harv|Caselles|Chambolle|Novaga|2007}}. A sensible extension of this model to colour images, called Colour TV, can be found in {{Harv|Blomgren|Chan|1998}}.